Why is it used with Dativ here:
"Wir gehen zusammen aus dem Haus."
"Wir gehen zusammen aus das Haus"?
Just like we should use Akkusativ in "Wir gehen in die Schule"(movement) and Dativ in "Wir sind in der Schule"(location).
This is because there are prepositions like
mit, nach, aus, zu, von, bei
that ALWAYS take Dative, even if this is indeed a movement in a direction. Another example with zu: (movement towards something):
Ich gehe zu dem Arzt
To make the above list more complete, the following prepositions are also only used with Dativ:
ab, außer, seit, entgegen, gegenüber.
The case is not only subject to the direction of motion, but rather to the preposition that ist used:
Wir gehen zu dem Haus
Wir gehen in das Haus
You need to know which prepositions require which case.
As I said elsewhere before--and though I'm not sure about the whole thing, it is anyway helpful here for illustrative purposes--try to think of "in das" as "in's", which it will be in the majority of cases in spoken German, and equate En "into" (compare "tooth" ~ "Zahn", read "inz", and note that the definite articles are an innovation that replaced earlier Germanic *sa; The difference may be merely prosodic), e.g. "in's Wasser gefallen". It is possible that "into" was an innovation as well, but that's besides the point.
Compare zur (seemingly "zu der"), "dar", En "there", etc. as well as the fixed expression "außer Haus" (outta house), and it should become obvious that the final consonants are morphemic, not mere contractions (they may be, but I don't wherefrom; "aus" for one can be reconstructed down to the Proto-Indo-European level with a dental consonant).
Pressumably, "aus dem Haus" had been ablative before, literally the case of coming from somewhere. This potentially fell in with "außer", whatever case that was. We see that some confusion arose, evident from "außerdem" ("apart from that, in addition to that)" which seems to mix cases. We see different endings in the same cases for different genders, or the same endings for different genders, and they do not agree accross the descendent languages from which Proto-Germanic is reconstructed.
So, in short, the answer is: because it's messed up is why. In long, there are good estimates of what it originally was (better than mine anyhow), but why it is what it is would involve a lot of guess work. I won't say that nominative had replaced ablativ, because nominative is primarily a case for subjects. Dativ is a real posibility, though, as far as I can tell. Compare "außen" (outside, outdoor), "draußen" (dar+außen, outside), vs "aus'em" (aus+dem). A few regional dialects do say "aus den".
I guess this stems from Latin as it is similar in Russian:
Static location goes with dative: im Wasser / aufm Wasser
Target movement goes with accusative: ins Wasser / aufs Wasser